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Complete Axiomatizations for Quantum Actions

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Abstract

We present two equivalent axiomatizations for a logic of quantum actions: one in terms of quantum transition systems, and the other in terms of quantum dynamic algebras. The main contribution of the paper is conceptual, offering a new view of quantum structures in terms of their underlying logical dynamics. We also prove Representation Theorems, showing these axiomatizations to be complete with respect to the natural Hilbert-space semantics. The advantages of this setting are many: (1) it provides a clear and intuitive dynamic-operational meaning to key postulates (e.g. Orthomodularity, Covering Law); (2) it reduces the complexity of the Solèr–Mayet axiomatization by replacing some of their key higher-order concepts (e.g. “automorphisms of the ortholattice”) by first-order objects (“actions”) in our structure; (3) it provides a link between traditional quantum logic and the needs of quantum computation.

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Authors and Affiliations

  1. Oxford University Computing Laboratory, Oxford University, Oxford, United Kingdom

    A. Baltag

  2. Vrije Universiteit Brussel, Brussel, Belgium

    S. Smets

  3. Vrije Universiteit Brussel, Flanders’ Fund for Scientific Research Post-Doc, Belgium

    S. Smets

Authors
  1. A. Baltag
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  2. S. Smets
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Corresponding author

Correspondence to S. Smets.

Additional information

PACS: 02.10.-v Logic; set theory and algebra; 03.65.-w Quantum mechanics; 03.65.Fd Algebraic methods; 03.67.-a Quantum information.

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Baltag, A., Smets, S. Complete Axiomatizations for Quantum Actions. Int J Theor Phys 44, 2267–2282 (2005). https://doi.org/10.1007/s10773-005-8022-2

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  • DOI: https://doi.org/10.1007/s10773-005-8022-2

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